The materials reviewed for Reveal Math Grade 5 meet expectations for developing conceptual understanding of key mathematical concepts, especially where called for in specific content standards or cluster headings.

The materials develop conceptual understanding throughout the grade level, with teacher guidance, through discussion questions and conceptual problems with low computational difficulty. Examples include:

• In Lesson 4-6, Represent Subtraction of Tenths and Hundredths, Explore and Develop, Develop the Math, the teacher is directed to “Make a false claim for students to critique. Write 0.04 - 0.01 = 0.03. Point to the equation and say This equation is correct. Yes or No? Ask students to correct the statement. Revisit this routine throughout the lesson to provide reinforcement.” This opportunity allows students to engage with their teacher in the conceptual development of 5.NBT.7, add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies...

• In Lesson 12-5, Solve Problems Involving Measurement Data on Line Plots, Guided Exploration, “What are the steps you would perform to solve this problem? Can you understand other students’ plans? How are their plans similar to yours? How are they different?” The teacher facilitates mathematical discourse and deepens conceptual understanding of 5.MD.2, make a line plot to display a data set of measurements in fractions of a unit. 

• In Lesson 13-4, Classify Triangles by Properties, Bring It Together, “How do you know if a triangle can be classified as a scalene, isosceles, or equilateral? What is similar about categories and subcategories in a hierarchy? What is different?” The teacher facilitates mathematical discourse and deepens conceptual understanding of 5.G.4, classify two- dimensional figures in a hierarchy based on properties.

The materials provide opportunities for students to independently demonstrate conceptual understanding through concrete, semi-concrete, verbal, and written representations. Examples include:

• In Lesson 2-1, Understand Volume, Activity Based Exploration, “demonstrate how to form rectangular prisms using the nets. Have students determine how many of each unit can fit inside the rectangular prism.” Students build conceptual understanding by using unit cubes, marbles, beans or other measurement units, 5.MD.3, recognize volume as an attribute of solid figures and understand concepts of volume measurement.

• In Lesson 2-4, Determine the Volume of Composite Figures, On My Own, Exercise 6, students “draw line(s) to show how you decomposed the figure. What is the volume of the figure?” This helps students build conceptual understanding of 5.MD.4, measure volumes by counting cubes, using cubic cm, cubic in, cubic ft, and improvised units.

• In Lesson 6-4, Represent Multiplication of Decimals, Differentiate, Differentiation Resource Book, Item 6, “Use an area model to solve. 6.2 x 2.1 = ___.” Students use concrete models or drawings to multiply decimals,  5.NBT.7, add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies...

• In Lesson 7-4, Represent Division of 2-Digit Divisors, On My Own, Exercise 1, “What is the quotient? Use an area model to solve.”Students use area models to independently demonstrate 5.NBT.6, find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value.

The materials reviewed for Reveal Math Grade 5 meet expectations that the materials develop procedural skills and fluency throughout the grade level. The materials provide opportunities for students to independently demonstrate procedural skills and fluency throughout the grade level. 

The materials develop procedural skill and fluency throughout the grade with teacher guidance, within standards and clusters that specifically relate to procedural skills and fluency, and build fluency from conceptual understanding. Examples include:

• Fluency Practice exercises are provided at the end of each unit. Each Fluency Practice includes Fluency Strategy, Fluency Flash, Fluency Check, and Fluency Talk. “Fluency practice helps students develop procedural fluency, that is, the ‘ability to apply procedures accurately, efficiently, and flexibly.’ Because there is no expectation of speed, students should not be timed when completing the practice activity.” Fluency Practice exercises in Grade 5 progress toward 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Lesson 3-1, Generalize Place Value, Number Routine: Where Does it Go?, “Students determine the location of a decimal on two number lines with different marked endpoints. Remind students that this is an estimation activity and exact locations are not needed.” Students build fluency of decimals, 5.NBT.3, read, write, and compare decimals to thousandths.

• In Lesson 5-4, Use Area Models to Multiply Multi-Digit Factors, Explore & Develop, Develop the Math, Activity-Based Exploration, students explore area models to determine different ways to decompose them to form partial products. “Ask students to write a multiplication problem using one 3-digit factor and a one 1-digit factor and draw an area model to represent the product. Have students record as many ways as possible to decompose the area model. Invite students to share ways they decomposed the area model, focus attention on similar methods of decomposing, such as decomposing by place value. ‘Do you think these methods of decomposing will work for multiplying two multi-digit numbers?’” This exploration provides an opportunity for students to develop procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm. 

• In Lesson 5-7, Multiply Multi-Digit Factors Fluently, Differentiate, Reinforce Understanding,  “Work with students in pairs using a spinner that contains 2-digit numbers. One student spins the spinner to obtain a factor. The other student rolls a number cube twice to produce a 2-digit factor. Help students multiply the two factors and then check their work using estimation. Have students repeat the process with new numbers.” The teacher works with students to build procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Unit 7, Divide Whole Numbers, Fluency Check, Fluency Talk, “Explain how you can use properties of operations to find the product of a number and a multiple of 10.” This discussion helps students build fluency of multiplication, 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

The materials provide opportunities for students to independently demonstrate procedural skill and fluency. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, Exit Ticket, Exercise 1, “Use a formula to find the volume of the rectangular prism,” Students independently demonstrate procedural  fluency of 5.MD.5, relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

• In Lesson 5-4, Use Area Models to Multiply Multi-Digit Factors, Differentiate, Building Proficiency, Student Practice Book, Item 3, students use area models and partial products to solve. “18 x 221 = _____.” Students independently demonstrate procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

• In Unit 14, Algebraic Thinking, Fluency Practice, Fluency Strategy, “You can choose a strategy to multiply. You can use an area model, partial products, or an algorithm.” Students practice different strategies to independently demonstrate procedural skill and fluency of 5.NBT.5, fluently multiply multi-digit whole numbers using the standard algorithm.

The materials reviewed for Reveal Math Grade 5 meet expectations for being designed so that teachers and students spend sufficient time working with engaging applications of the mathematics. Additionally, the materials provide students with the opportunity to independently demonstrate multiple routine and non-routine applications of the mathematics throughout the grade level. 

The materials provide specific opportunities within each unit for students to engage with both routine and non-routine application problems. In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Focus, Coherence, Rigor, Application, “Students encounter real-world problems throughout each lesson. The On My Own exercises include rich, application-based question types, such as ‘Find the Error’ and ‘Extend Thinking.’ Daily differentiation provides opportunities for application through the Application Station Cards, STEM Adventures, and WebSketch Explorations. The unit performance task found in the Student Edition offers another opportunity for students to solve non-routine application problems.” 

The materials develop application throughout the grade as students solve routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, Practice & Reflect, On My Own, Problem 8, “A freezer, shaped like a rectangular prism, is 6 feet long, 2 feet wide, and 3 feet tall. What is the volume of this freezer.”This exercise allows students to develop and apply mathematics of 5.MD.5b, relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

• In Lesson 8-1, Division Patterns with Decimals and Powers of 10, On My Own, Exercise 10, “Danny walks 567.3 miles in 100 days. Michelle walks a 567.3 miles by walking 0.1 miles each day. Who walked for more days? Who walked farther each day? Explain.” This exercise allows students to develop and apply mathematics of 5.NBT.2, explain patterns in the placement of the decimal point when the decimal is multiplied or divided by a power of 10.

• In Lesson 11-2, Solve Problems Involving Division, Additional Practice, Exercise 3, “A 10-kilometer race is divided into 3 equal sections. How long is each section of the race?” This problem allows students to apply mathematics  of 5.NF.3, interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers.

The materials develop application throughout the grade as students solve non-routine problems in a variety of contexts, and model the contexts mathematically within standards and clusters that specifically relate to application, both dependently and independently. Examples include:

• In Lesson 3-5, Use Place Value to Round Decimals, Extend Thinking, Item 3, “Quentin drives 632.074 miles from Sacramento, California to Las Vegas, Nevada one day and then drives 632.32 miles from Las Vegas to Santa Fe, New Mexico the Next Day. If the distance Quentin traveled on the first day was rounded to 632.32, what is a possible distance he could have traveled on that day?” This is a non-routine problem because students can answer with any number less than 632.325 and equal to or greater than 632.315. This exercise allows students to develop and apply mathematics of 5.NBT.4, use place value understanding to round decimals to any place.

• In Unit 10, Multiply Fractions, Application Station, Connection Card, Fraction of a Fraction, “Sometimes, a recipe makes much more food than you need. Find a recipe that uses healthy ingredients and makes a lot of food. Be sure to choose a recipe that lists the amount of each ingredient as a fraction. Next, create three different stories for why you are only making $\frac{1}{5}$, $\frac{2}{3}$ , and $\frac{7}{8}$ of the recipe. Draw area models to represent $\frac{1}{5}$, $\frac{2}{3}$ , and $\frac{7}{8}$ of the amount of each ingredient needed. 1. Explain how you determined the side lengths of each area model? 2. Explain how you know whether you are making less, the same amount, or more food than the original recipe? 3. Compare the different amounts of each ingredient after determining $\frac{1}{5}$, $\frac{2}{3}$ , and $\frac{7}{8}$ of each. This exercise allows students to develop and apply mathematics of 5.NF.6, solve real world problems involving multiplication of fractions and mixed numbers. 

• In Unit 14, Algebraic Thinking, Application Station, Real World Card, Earning an Income, “People are paid money they earn for performing a job, either as an hourly wage or a salary. An hourly wage means getting paid the same amount for every hour that you work. But a salary is a fixed amount of money divided up over the year and usually paid every week or two. Choose 5 jobs that interest you. Research and record the starting income and whether it is an hourly wage or a yearly salary. Determine the amount of money earned for each 40−hour week. Discuss your strategies for how you will determine this with your group. This material may be reproduced for licensed classroom use only and may not be further reproduced or distributed. Then, create 5 tables, one for each job, so that the first column is the number of weeks and the second column is the amount of money earned each week. Describe the relationship between the terms in each column as you extend the table. Write the corresponding terms as ordered pairs, plot the ordered pairs on a coordinate plane, and connect the points. Finalize each by describing the relationship between the corresponding terms in the table. 1. What are some other ways, besides income, that employers attract employees to work for them? 2. What are some reasons for choosing one job over another? 3. What are some reasons for choosing a job that might pay less?” This exercise allows students to develop and apply mathematics of 5.NBT.7, apply and extend previous understandings of division to divide unit fractions by whole numbers by unit fractions.

The materials reviewed for Reveal Math Grade 5 meet expectations in that the three aspects of rigor are not always treated together, and are not always treated separately. There is a balance of the three aspects of rigor within the grade. Additionally, multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout each grade level. 

All three aspects of rigor (conceptual understanding, procedural skill & fluency, and application) are present independently throughout the grade level. Examples include:

• In Lesson 2-1, Understand Volume, On My Own, Work Together, students develop conceptual understanding of volume as the amount of space taken up by a solid object. “One student used marbles to pack a rectangular prism. Another student used unit cubes. What do you notice about these strategies?”

• In Lesson 4-1, Estimate Sums and Differences of Decimals, On My Own, Problems 1-8, students build fluency with place-value concepts and learn procedures for estimating sums and differences of decimals. For example, Problem 1, “9.86 + 4.30.” Problem 3, “3.92 + 6.14.” Problem 5, “8.32 - 5.9.”

• In Lesson 9-9, Solve Problems Involving Fractions and Mixed Numbers, On My Own, Problem 3, students add and subtract mixed numbers involving unlike denominators to solve real world problems. “Alyana buys $4\frac{3}{10}$ pounds of potatoes. She uses $2\frac{3}{4}$ pounds in a recipe. How many pounds does she have left?

The materials provide a balance of the three aspects of rigor as multiple aspects of rigor are engaged simultaneously to develop students’ mathematical understanding of a single topic/unit of study throughout the grade level. Examples include:

• In Lesson 2-3, Use Formulas to Determine Volume, On My Own, Problem 9, students use their conceptual understanding of volume to develop the formula to calculate volume of rectangular prisms, and apply the formula to solve real-world problems. “An Olympic swimming pool is 2 meters deep. What is the volume of the swimming pool?” A pool with dimensions of 50m by 25m is shown.

• In Lesson 6-2, Estimate Products of Decimals, On My Own, Problems 7-12, students extend their conceptual understanding of estimation to build procedural skill and fluency of estimating products of decimals. Problem 7, “Estimate each product by finding a range. Show your work. 4.93 x 7.88.” Problem 10, “Estimate each product by finding a range. Show your work. 4.1 x 13.5.” Problem 12, “Estimate each product by finding a range. Show your work. 16.12 x 3.55.”

• In Lesson 12-3, Solve Multi-Step Problems Involving Measurement Units, On My Own, Problem 4, students build their procedural skill and fluency with multiplication involving whole numbers and fractions to solve real-world problems involving measurement conversions. “A track at the school is 400 meters long. Jackson walks around the track  $3\frac{1}{2}$ times. How many kilometers did Jackson walk?

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP1: Make sense of problems and persevere in solving them; and MP2: Reason abstractly and quantitatively, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is labeled as MPP Reason abstractly and quantitatively, rather than MP1 or MP 2. Within each of the lesson components, mathematical practices are not labeled or identified, leaving where they are specifically addressed up for interpretation and possible misidentification.

The materials provide intentional development of MP1: Make sense of problems and persevere in solving them, in connection to grade-level content. Examples include: 

• In Lesson 6-6, Explain Strategies to Multiply Decimals, Differentiate, Extend Thinking, Differentiation Resource Book, Exercise 4, “Numbers 1-6 are solutions to multiplication problems. 4.  24.30 _____ and _____. Match each problem A-L to its solution above in 1 to 6.” Students engage with MP1 as they make connections between equations, pictorial representations, and word problems.

• In Lesson 9-8, Add and Subtract mixed Numbers with Regrouping, On My Own, Reflect, “When is regrouping necessary when adding and subtracting mixed numbers?” Students engage in MP1 as they reflect on problem solving strategies.

• In Unit 11, Divide Fractions, Unit Review, Exercise 12, What equation does this model most likely represent? A. 5 $\div$ 3 = n, B. 3 $\div$ $\frac{1}{5}$ =  n, C. 5 $\div$ $<em>\frac{</em>1}{3}$ = n, D. 3 $\div$ 5 = n”  Students engage with MP1 as they make sense of a model and match it to the corresponding equation. 

The materials provide intentional development of MP2: Reason abstractly and quantitatively, in connection to grade-level content. Examples include:

• In Teacher’s Guide, Lesson 3-4, Compare Decimals, Guided Exploration, “Students extend their understanding of comparing whole numbers using place value to decimal numbers. What different ways can you write the comparison statement? How are they the same? How are they different? Explain your reasoning. Think About it: Are there other models or tools you could use to compare decimal numbers? How could writing them in expanded form help?” Students engage with MP2 as they explain/discuss what the numbers or symbols in an expression/equation represent. 

• In Lesson 7-1, Division Patterns with Multi-Digit Numbers, Differentiate, Build Proficiency, Student Practice Book, Exercise 11, “There are 32,000 quarters in rolls of 40, how many rolls of quarters are there?” Students engage with MP2 as they consider units involved and attend to the meaning of quantities.

In Lesson 11-5, Represent Division of Unit Fractions by Non-Zero Whole Numbers, Explore & Develop, Learn, Work Together, “Peter has $\frac{1}{4}$ gallon of water. He equally shares the water among his 2 dogs. How much water will each dog get?” Students engage with MP2 as they consider units involved and attend to the meaning of quantities.

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP3: Construct viable arguments and critique the reasoning of others, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both of these sections, the mathematical practice is labeled MPP: Construct viable arguments and critique the reasoning of others, rather than MP3 Construct viable arguments and critique the reasoning of others. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

Examples of intentional development of students constructing viable arguments in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Unit 2 , Volume of Rectangular Prisms, Math Probe, Exercise 1, students construct viable arguments as they determine which expression(s) can be selected to determine the volume. “Which expression(s) can be used to determine the volume of the rectangular prism shown. Select all that apply. Do not actually find the volume of the prism. Explain your choice(s).”

• In Unit 3, Unit Review, Performance Task, Part B, students construct viable arguments as they compare two decimals to thousandths. “Jupiter has 67 confirmed moons. Each moon orbits at different speeds. One moon takes 259.22 Earth days to orbit Jupiter and another one takes 259.653 Earth days. Use >, <, or = to compare the orbit speeds. Explain your answer.”

• In Lesson 13-5, Properties of Quadrilaterals, Practice & Reflect, On My Own, Exercise 13, students construct viable arguments as they classify two-dimensional figures in a hierarchy based on properties. “How are all quadrilaterals the same? How are they different?”

Examples of intentional development of students critiquing the reasoning of others in connection to grade-level content, including guidance for teachers to engage students in MP3 include:

• In Lesson 3-3, Read and Write Decimals, Differentiate, Additional Practice, Exercise 6, students critique the reasoning of others. “Colby says that $\frac{27}{100}$ written in word form is twenty-seven thousandths. Do you agree? Explain?”

• In Lesson 5.2, Patterns When Multiplying a Whole Number by Powers of 10, Differentiate, Student Practice Book, Exercise 13, students critique the reasoning of others as they explain patterns in the number of zeros of the product when multiplying a number by powers of 10. “Herschel thinks that 30 x 1,000 = 30,000. How would you respond to Herschel?”

• In Lesson 11-1, Relate Fractions to Division, Own My Own, Exercise 12, students critique the reasoning of others. “Spencer divides 6 pounds of food from the food drive into 3 boxes. He says each box has $\frac{3}{6}$ pounds of food. Is he right? How do you know?”

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP4: Model with mathematics; and MP5: Choose tools strategically, for students, in connection to the grade-level content standards, as expected by the mathematical practice standards. 

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, in both sections, the mathematical practice is MPP Model with mathematics, rather than MP4. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students modeling with mathematics in connection to grade-level content, including guidance for teachers to engage students in MP4 include:

• In Lesson 2-2, Use Unit Cubes to Determine Volume, Reinforce Understanding, Small Group, “Give each student 30 unit cubes. Have each student create a rectangular solid using some or all of the cubes to find the volume of the figure. Have students switch figures with another student and find the volume of the figure. Then have students switch again so that each student finds the volume of all three figures. If necessary, remind students that they can count the number of cubes used to find the volume.” Students engage with MP4 as they check to see whether an answer makes sense and change the model when necessary as they measure volumes by counting unit cubes. 

• In Lesson 2-3, Use Formulas to Determine Volume, Differentiate, Build Proficiency, Student Practice Book, Exercise 7, “A window air conditioner can cool a space of up to 50 cubic meters. The floor of a room has an area of 16 square meters, and the height of the walls is 3 meters. Will the air conditioner be able to cool the room? Explain.” Students engage in MP4 as they use the math they know to solve problems and everyday situations.

• In Lesson 8-3, Represent Division of Decimals by a Whole Number, Guided Exploration, Math is...Modeling, students answer, “How do decimal grids help you understand dividing decimals by a whole number?” Students reflect on using decimal grids as a strategy to help them divide decimals by whole numbers.

Examples of intentional development of students using appropriate tools strategically in connection to grade-level content, including guidance for teachers to engage students in MP5 include:

• In Lesson 4-1, Estimate Sums and Differences of Decimals, Practice & Reflect, On My Own, Exercise 8,  “What is a reasonable estimate for the sum or difference? Explain the strategy you used? 5.42 - 1.7=   .” Students engage in MP5 as they choose an appropriate strategy to make a reasonable estimate.

• In Unit 9, Add and Subtract Fractions, Unit Resources, Application Station, Real World Card, Create and Solve, “Create a multi-step problem that adds and subtracts mixed numbers to solve. Then use a digital tool to present the problem to your group and find the solution together. 1. What digital tool did you use to present your problem? 2. What did you like about this digital tool? 3. Was there something you could not do using this digital tool? 4. How might you use this digital tool again?” Students engage in MP5 to choose tools (and create a digital tool) to solve multi-step word problems.

• In Lesson 10-3, Multiply Mixed Numbers, Assess, Exit Ticket, Exercise 3, “Jacob chooses a pumpkin that weighs $6\frac{3}{5}$ kilograms. Kaleigh chooses a pumpkin that weighs $1\frac{3}{4}$ times as much as Jacob’s pumpkin. How many kilograms does Kaleigh’s pumpkin weigh?” Students engage in MP5 as they choose tools and strategies to solve fraction word problems.

The materials reviewed for Reveal Math Grade 5 meet expectations that there is intentional development of MP6: Attend to precision; and attend to the specialized language of mathematics for students in connection to the grade-level content standards, as expected by the mathematical practice standards.  

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Attend to precision, rather than MP6: Attend to precision. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification.

The instructional materials address MP6 in the following components:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Language of Math, Unit-level Features, “The Language of Math feature highlights math terms that students will use during the unit. New terms are highlighted in yellow. Terms that have a math meaning different from everyday means are also explained.” Math Language Development, “This feature targets one of four language skills - reading, writing, listening, speaking - and offers suggestions for helping students build proficiency with these skills in the math classroom.” Lesson Level Features, “The Language of Math feature promotes the development of key vocabulary terms that support how we talk about and think about math in the context of the lesson content.” Each Unit Review also includes a vocabulary review component which references specific lessons within the unit.

Examples of intentional development of MP6: attend to precision, in connection to the grade-level content standards, as expected by the mathematical practice standards, including guidance for teachers to engage students in MP6 include:

• In Lesson 5-7, Multiply Multi-Digit Factors Fluently, Reinforce Understanding, Independent Work, Exercise 8, “Find the product of each equation using an algorithm. 1,786 x 62.” Students attend to precision by multiplying a four-digit whole number by a two-digit whole number with accuracy.

• In Lesson 6-1, Patterns When Multiplying Decimals by Powers of 10, On My Own, Problem 6, “Juan walks 4.7 x 10³ meters from his house to the museum. Mary walks 9.3 x 10² meters from her house to the museum. Who walks farther, Juan or Mary? How do you know?” Students attend to precision in calculations with exponents.

• In Unit 8, Divide Decimals, Performance Task Exercise 11, “What is the quotient? 9.72 $\div$3” Students attend to precision as they divide decimals.

Examples of where the instructional materials attend to the specialized language of mathematics, including guidance for teachers to engage students in MP6 include:

• In Unit 3, Teacher Edition, Math Probe, Take Action, “Build place-value ideas by using language that reinforces place value. For example, rather than reading 3.45 as three point four five, students should read it as three and forty-five hundredths.” Students attend to the specialized language of math.

• In Lesson 7-1, Division Patterns with Multi-Digit Numbers, Own My Own, Reflect, “How does using place-value patterns and basic facts help you divide whole numbers by multiples of 10?” Students attend to the specialized language of math including place-value, division, and multiples.

• In Unit 9, Add and Subtract Fractions, Unit Review, Vocabulary Review, Exercise 2, “In order to find the sum of fractions with unlike denominators, you can rewrite the fraction using ______ so that the denominators are alike.” Students attend to the specialized language of math by completing a vocabulary review to check their understanding of fractions.

The materials reviewed for Reveal Math Grade 5 meet expectations for supporting the intentional development of MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning, for students, in connection to grade-level content standards, as expected by the mathematical practice standards.

Each Unit Overview, Math Practices and Processes section, identifies one mathematical practice that is prevalent in the unit, and gives an overview of its use within the unit. In the Standards section of each lesson, mathematical practices for the lesson are also identified; however, the mathematical practice is labeled MPP Look for and make use of structure, rather than MP7: Look for and make use of structure; and MP8: Look for and express regularity in repeated reasoning. Additionally, the math practices are not identified within the lesson sections, therefore leaving where they are specifically addressed up for interpretation and possible misidentification. 

Examples of intentional development of students looking for and making use of structure, to meet its full intent in connection to grade-level content, including guidance for teachers to engage students in MP7 include:

• In Lesson 6-1, Patterns When Multiplying Decimals by Powers of 10, Reinforce Understanding, Exercise 5, “Use patterns to help you find the value of each expression. 1.3 x 10² = ____, 1.3 x 10³ = ____, 1.3 x 10⁴ = ____.” Students engage in MP7 as they make connections between multiplying whole numbers by powers of ten to multiplying decimals by powers of ten.

• In Lesson 9-8, Launch, Notice and Wonder, “What do you notice? What do you wonder? Pose Purposeful Questions: The questions that follow may be asked in any order. They are meant to help advance students’ exploration of using regrouping to add and subtract mixed numbers and are based on possible comments and questions that students may make during the share out. What is missing from each part? How can you make wholes? Let’s think about when we need to, and how we can, use regrouping to subtract mixed numbers.”  Students engage in MP7 as they look for and explain the structure within mathematical representations. 

• In Lesson 13-5, Properties of Quadrilaterals, Guided Exploration, Math is...Structure, students answer “How can you compare the attributes of quadrilaterals and triangles?” Students engage with MP7 as they determine the properties that define categories.

Examples of intentional development of students looking for and expressing regularity in repeated reasoning, including guidance for teachers to engage students in MP 8 include:

• In Lesson 5-6, Relate Partial Products to an Algorithm, On My Own, Reflect, “How are partial products and an algorithm for multiplication related?” Students engage with MP8 as they make generalizations about multiplication strategies.

• In Lesson 11-4, Divide whole Numbers by Unit Fractions, On My Own, Exercise 15, Extend Your Thinking, “When a whole number is divided by a fraction that is less than 1, will the quotient always be greater than the whole number? Explain why or why not.” Students engage with MP8 as they evaluate the reasonableness of the answers and thinking.

• In Lesson 14-5 Relate Numerical Patterns, Guided Exploration, Math is...Structure, students answer “How are the terms in Pattern A related to their corresponding terms in Pattern B?” Students engage with MP8 as they make inferences about inverse relationships.

The materials reviewed for Reveal Math Grade 5 meet expectations for providing teacher guidance with useful annotations and suggestions for how to enact the student materials and ancillary materials, with specific attention to engaging students in order to guide their mathematical development.

Materials provide comprehensive guidance that will assist teachers in presenting the student and ancillary materials. Examples include:

• The Implementation Guide provides a program guide, which includes a program overview, the program components, unit features, instructional model, lesson walk-through, and a brief description of the different unit components, such as Math is…, focus, coherence, rigor, and language of math.  

• The Implementation Guide provides pacing for each unit; mapping out the lessons in each unit and how many days the unit will take.

• The Unit Planner contains an overview of the Lessons within the unit, Math Objective, Language Objective, Key Vocabulary, Materials to Gather, Rigor Focus, and Standard.

• The Unit Overview provides a description for teachers as to how the unit connects to Focus, Coherence, and Rigor. 

• Within each lesson, the Language of Math section, provides teachers with specific information about the vocabulary used in lessons and how to utilize vocabulary cards to enhance learning experiences. 

• In Unit 2, Volume, Unit Overview, Effective Teaching Practices, Elicit and Use Evidence of Student Thinking, “Look for evidence of student thinking and evaluate their growth toward conceptual understanding. Before, during, and after learning a new skill or concept, students should be assessed to see if they are understanding the new information or if they have any misconceptions of past information...Assessment is continuous because students’ understanding drives instruction. Sometimes topics that have been previously covered need to be approached in a different way because students may be struggling with a prior topic that is stopping future learning from occurring.”

• In Unit 6, Multiply Decimals, Unit Overview, Math Practices and Processes, Attend to Precision, “To help students develop the habit of attending to precision, assign tasks that require precision and set clear expectations. For example:

  • Have students talk about their representations with area models and decimal grids.

  • Have students discuss how they estimate products, how they know their estimates are reasonable, and how far away a reasonable answer could be from an estimate.

  • Have students explain the thought process they use to label their area representations.”

  • Pay attention to whether and how students attend to the units in the problems they solve and ask them questions that lead to thinking about units.

  • Have students describe the pattern they discovered in the position of the decimal point in products.”

Materials include sufficient and useful annotations and suggestions that are presented within the context of the specific learning objectives. The materials provide information about planning instruction, and give suggestions for presenting instructional strategies and content, as well as mathematical practices. Examples include:

• In Lesson 3-3, Read and Write Decimals, Notice & Wonder, Teaching Tip, “Encourage students to add onto another student's idea. This promotes opportunities for participation from a variety of students. You can ask questions, such as Would someone like to add on? to help elicit more discussion when few students are talking.”

• In Lesson 4-6, Represent Subtraction of Tenths and Hundredths, English Learner Scaffolds, “Entering/Emerging Support students in understanding lengths. Put three pencils on the desk, different sizes. Say Let’s find the lengths of the pencils. Measure each pencil and say The length is (5 inches) for each. Repeat with new objects. Finally, show students three more objects and ask them What are the lengths? Prompt students to measure each object and say the lengths aloud.”

The materials reviewed for Reveal Math Grade 5 meet expectations for including standards correlation information that explains the role of the standards in the context of the overall series.

Correlation information is present for the mathematics standards addressed throughout the grade level. Examples of how individual units, lessons, or activities throughout the series are correlated to the CCSSM include:

• In the Digital Teacher Center, Program Overview: Learning & Support Resources, Implementation Guide, Correlations, identifies the standards included in each lesson. This guide also indicates whether the standards are considered major, supporting, or additional standards. 

• Each Unit Planner includes a pacing guide identifying the standards that will be addressed in each lesson.

• In Lesson 2-2, Use Unit Cubes to Determine Volume, the materials identify focus standards 5.MD.3, recognize volume as an attribute of solid figures and understand concepts of volume measurement, and 5.MD.4, measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. The lesson also identifies MP7. 

• In Lesson 8-3, Represent Division of Decimals by a Whole Number, the materials identify focus standard 5.NBT.7, add, subtract, multiply, and divide decimals to hundredths. The lesson also identifies the MPs 4 and 8. 

Explanations of the role of the specific grade-level mathematics are present in the context of the series, and teacher materials provide information to allow for coherence across multiple course levels. This allows the teacher to make prior connections and teach for connections to future content. Examples include:

• The Unit Overview includes the section, Coherence, identifying What Students Have Learned, What Students Are Learning, What Students Will Learn. In Unit 3, Place Value and Number Relationships, What Students Have Learned, “Whole Number Place Value Students recognized that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. (Grade 4), Volume Students understood volume. (Grade 2)” In What Students Are Learning, “Decimal Place Value Students understand decimal place value., Reading and Writing Decimals Students read and write decimals in number, word, and expanded form., Comparing Decimals Students compare decimals using the same strategies used for whole numbers., Rounding Decimals Students round decimals using the same strategies used for whole numbers.” In What Students Will Learn, “Add and Subtract Decimals Students will add and subtract decimals. (Unit 4), Add, Subtract, Multiply, and Divide Multi-Digit Decimals Students will fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (Grade 6)”

• Each lesson begins by listing the standards covered within the lesson, indicates whether the standard is a major, supporting or additional standard and identifies the Standards for Mathematical Practice for the lesson. Each lesson overview contains a coherence section that provides connections to prior and future work. In Lesson 4-1, Estimate Sums and Differences of Decimals, Coherence, Previous, “Students fluently added and subtracted multi-digit whole numbers using the standard algorithm (Grade 4)., Students generalized their understanding of place value in decimals (Unit 3).” Now, “Students use place-value strategies to estimate sums and differences of decimals. Students describe and explain estimation strategies.” Next, “Students use representations to add with decimals and explain their strategies (Unit 4)., Students fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm, (Grade 6).”

The materials reviewed for Reveal Math Grade 5 meet expectations for providing explanations of the instructional approaches of the program and identification of the research-based strategies.

The materials explain the instructional approaches of the program. Examples include:

• Digital Teacher Center, Program Overview: Learning & Support Resources, Teacher Welcome Letter Template specifies “Reveal Math, a balanced elementary math program, develops the problem solvers of tomorrow by incorporating both inquiry-focused and teacher-guided instructional strategies within each lesson.” 

• Teacher Guide, Volume 1, Welcome to Reveal Math, the overall organization of the math curriculum has five goals:

  • “The lesson model offers two instructional options for each lesson: a guided exploration that is teacher-guided and an activity-based exploration that has students exploring concepts through small group activities and drawing generalizations and understanding from the activities.

  • The lesson model incorporates an initial sense-making activity that builds students’ proficiency with problem solving. By focusing systematically on sense-making, students develop and refine not just their observation and questioning skills, but the foundation for mathematical modeling.

  • Both instructional options focus on fostering mathematical language and rich mathematical discourse by including probing questions and prompts.

  • The Math is… unit builds student agency for mathematics. Students consider their strengths in mathematics, the thinking habits of proficient “doers of mathematics,” and the classroom norms that are important to a productive learning environment.

  • The scope and sequence reflects the learning progressions recommended by leading mathematicians and mathematics educators. It emphasizes developing deep understanding of the grade-level concepts and fluency with skills, while also providing rich opportunities to apply concepts to solve problems.”

The Implementation Guide, located in the Digital Teacher Center, further explains the instructional approaches of specific components of the program. Examples include: 

• Unit Features, Unit Planner, “Provides at-a-glance information to help teachers prepare for the unit. Includes pacing: content, language, and SEL objectives; key vocabulary including math and academic terms; materials to gather; rigor focus; and standard (s).”

• Unit Features, Readiness Diagnostic, “Offers teachers a unit diagnostic that can be administered in print or in digital. The digital assessment is auto-scored. Assesses prerequisite skills that students need to be successful with unit content. Item analysis lists DOK level, skill focus, and standard of each item. Item analysis also lists intervention lessons that teachers can assign to students or use in small group instruction.”

• Unit Features, Spark Student Curiosity Through Ignite! Activities, “Each unit opens with an Ignite! Activity, an interesting problem or puzzle that: Sparks students’ interest and curiosity, Provides only enough information to open up students’ thinking, and Motivates them to persevere through challenges involved in problem solving.”

• Instructional Model, “Reveal Math’s lesson model keeps sense-making and exploration at the heart of learning. Every lesson provides two instructional options to develop the math content and tailor the lesson to the needs and structures of the classroom.” Each lesson follows the same structure of a “Launch, Explore & Develop, Practice & Reflect, Assess and Differentiate.” Each of these sections is further explained in the instruction manual.

• Number Routines, in each lesson there is a highlighted number routine for teachers to engage students with. These routines “are designed to build students’ proficiency with number and number sense. They promote an efficient and flexible application of strategies to solve unknown problems…”

The Implementation Guide, located in the Digital Teacher Center, discusses some of the research based features of the program. Examples include: 

• Implementation Guide, Effective Mathematical Teaching Practices, “Reveal Math’s instructional design integrates the Effective Mathematics Teaching Practices from the National Council of Teachers of Mathematics (NCTM). These research-based teaching practices were first presented and described in NCTM’s 2014 work Principles to Action: Ensuring Mathematical Success for All.”

• Implementation Guide, Social and Emotional Learning, “In addition to academic skills, schools are also a primary place for students to build social skills. When students learn to manage their emotions and behaviors and to interact productively with classmates, they are more likely to achieve academic success Research has shown that a focus on helping students develop social and emotional skills improves not just academic achievement, but students’ attitudes toward school and prosocial behaviors (Durlak et al., 2011)...”

• Implementation Guide, Support for English Learners, Lesson-level support, English Learner Scaffolds, each lesson has an “English Learner Scaffolds” section to support teachers with “scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, Developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language Development Standards.”

• Implementation Guide, Math Language Routines, throughout the materials certain language routines are highlighted for teachers to encourage during a lesson, these routines were developed by a team of authors at Center for Assessment, Learning and Equity at Standard University and are “based on principles for the design of mathematics curricula that promote both content and language.” In the implementation guide, the material lists all eight Math Language routines and their purposes, “MLR1: Stronger and Clearer Each Time - Students revise and refine their ideas as well as their verbal or written outputs.”

• Implementation Guide, Math Probe - Formative Assessment, each unit contains a Math Probe written by Cheryl Tobey. Math Probes take time to discover what misconceptions might still exist for students. Designed to ACT, “The teacher support materials that accompany the Math Probes are designed around an ACT cycle - Analyze the Probe, Collect and Assess Student Work, and Take Action. The ACT cycle was originally developed during the creation of a set of math probes and teacher resources for a Mathematics and science Partnership Project.”

The materials reviewed for Reveal Math Grade 5 meet expectations for providing a comprehensive list of supplies needed to support instructional activities.

The Digital Teacher Center, Program Resources: Course Materials, Planning Resources, Materials List: Grade 5 specifies the comprehensive materials list. The document specifies classroom materials (e.g., index cards, plastic straws, spaghetti noodles, etc.), materials from a manipulative kit (e.g., geoboards, base-ten blocks, transparent spinner, etc.), non-consumable teaching resources (e.g., pattern blocks, blank number lines, decimal cards, etc.), and consumable teaching resources (e.g., blank partial quotients, venn diagrams, coordinate planes, etc.).

In the Teacher Edition, each Unit Planner page lists materials needed for each lesson in the unit, for example, Unit 8, Divide Decimals, Materials to Gather:

• “Lesson 8-1 - base-ten blocks, calculators, hundred grids

• Lesson 8-2 - calculator, number cubes

• Lesson 8-3 - bills and coins manipulatives, index cards, Tenths and Hundredths Representations Teaching Resource

• Lesson 8-4 - number cubes

• Lesson 8-5 - 10 x 10 Grids Teaching Resource

• Lesson 8-6 - Tenths and Hundredths Representations.”

At the beginning of each lesson in the “Materials” section, a list of materials needed for each part of the lesson is provided:

• In Lesson 2-1, Understand Volume, Materials, “The materials may be for any part of the lesson: Nets Teaching Resource, centimeter cubes, marbles, beans, or other measurement units.”

• In Lesson 4-5, Represent Subtraction of Decimals, Materials, “The materials may be for any part of the lesson: Blank Number Lines Teaching Resource, number cubes, Tenths and Hundredths Teaching Resource.”

• In Lesson 5-7, Multiply Multi-Digit Factors Fluently, Materials, “The materials may be for any part of the lesson: Multiplication Algorithm Teaching Resource, number cubes, spinners.”

The materials reviewed for Reveal Math Grade 5 meet expectations for including an assessment system that provides multiple opportunities throughout the grade, course, and/or series to determine students' learning and sufficient guidance to teachers for interpreting student performance and suggestions for follow-up.

Each unit, beginning with Unit 2, offers a Readiness Diagnostic, that assesses the content of the unit and gives teachers a snapshot of the prerequisite skills the students already possess. Each unit also includes a Unit Assessment that evaluates students’ understanding of and fluency with concepts and skills from the unit. In the Teacher Edition, an Item Analysis lists each item’s DOK level, skill focus, content standard, and a Guided Support Intervention Lesson that teachers can assign or use for small groups or remediation. For example:

• In Unit 6, Multiply Decimals, Unit Assessment (Forms A and B), Item 3 lists “Estimate Products (Decimal Number Factors)” as the Guided Support Intervention Lesson. This resource can be located in the Digital Teacher Center in the Targeted Intervention section of the Unit.

Unit Performance Tasks include a scoring rubric that evaluates student work for each section on a 2, 1, or 0 point scale. No follow-up guidance is provided for the Performance Task. For example:

• In Unit 3, Place Value and Number Relationships, Performance Task Part C, Rubric (2 points), “2 Points: Student’s work identifies correct range of place values for rounding decimals. The student’s explanation is reasonable. 1 Point: Student’s either identifies correct range of place values for rounding decimals or has a reasonable explanation. 0 Points: Student’s does not identify the correct range of place values for rounding decimals. The student’s explanation is not reasonable.”

Math Probes analyze students’ misconceptions and are provided at least one time per Unit, beginning with Unit 2. In the Teacher Edition, “Authentic Student Work” samples are provided with correct student work and explanations. An “IF incorrect…, THEN the student likely…Sample Misconceptions” chart is provided to help teachers analyze student responses. A Take Action section gives teachers suggestions and resources for follow up or remediation as needed. There is a “Revisit the Probe” with discussion questions for students to review their initial answers after they are provided additional instruction, along with a Metacognitive Check for students to reflect on their own learning. For example:

• In Unit 10, Multiply Fractions, Math Probe Fraction Problems, Analyze The Probe, Targeted Concept, “Determine the product of a whole number and a fraction by reasoning about the magnitude of numbers and the meaning of multiplication within a word-problem context.” Students “Choose the best estimate for each problem. Do not actually solve the problems.” For example, “Problem 3, Chantal drinks $\frac{2}{3}$ cup of orange juice every morning. How many cups does she drink in 20 days? Circle the best estimate. A. 6 cups, b. 10 cups, c. 14 cups, d. 19 cups.” Targeted Misconceptions: Some students choose an incorrect operation when solving problems. They may misinterpret the problem context or focus on “key words” that may suggest an incorrect operation. Some students misinterpret an expression for an equation.” Sample Student work is provided, along with “IF incorrect...THEN the student likely…” explanations of the sample misconception are provided.

Exit Tickets are provided at the end of each lesson and evaluate students’ understanding of the lesson concepts and provide data to inform differentiation. Each includes a Metacognitive Check allowing students to reflect on their understanding of lesson concepts on a scale of 1 to 3, with 3 being the highest confidence, and beginning in Unit 2, include an Exit Skill Tracker that lists each item’s DOK, skill, and standard. The Exit Ticket Recommendations chart provides information regarding which differentiation activity to assign based on the student’s score. For example, “If students score…Then have students do” which provides teachers information on what Differentiation activities to use such as Reinforce Understanding, Build Proficiency or Extend Thinking. For example:

• In Lesson 12-1, Measurement and Data, Exit Ticket, Item 4, “Ginny jumped 6 feet. How many yards did Ginny jump? A. 2 yards, b. 3 yards, c. 12 yards, d. 18 yards.” Exit Ticket Recommendations: “If students score 4 of 4, Then have students do Additional Practice or any of the B (Build Proficiency) or E (Extend Thinking) activities. If students score 3 of 4, Then have students Take Another Look or any of the B activities. If students score 2 or fewer of 4, Then have students do Small Group Intervention or any of the R (Reinforce Understanding) activities.”

The materials reviewed for Reveal Math Grade 5 meet expectations that assessments include opportunities for students to demonstrate the full intent of grade-level standards and practices across the series. 

Reveal Math offers a variety of opportunities for students to demonstrate the full intent of grade-level standards and mathematical practices. While content standards and DOK levels are consistently identified for teachers in the Teacher Edition, and content standards are labeled for students in digital assessments, the standards for mathematical practice are not identified for teachers or students. It was noted that although assessment items do not clearly label the MPs, students are provided opportunities to engage with the mathematical practices.

Unit Readiness Diagnostics are given at the beginning of each unit, beginning with Unit 2. Formative assessments include Work Together, Exit Tickets, and Math Probes. Summative assessments include Unit Assessment Forms A and B, and Unit Performance Tasks at the end of a unit. Benchmark Assessments are administered after multiple units, and an End of the Year (Summative) Assessment is given at the end of the school year. Examples include:

• In Lesson 2-5, Solve Problems Involving Volume, Assess, Exit Ticket, Item 2, supports the full intent 5.MD.5 (Relate Volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume), and MP1 (Make sense of problems and persevere in solving them) as students determine which equation can be used to find the width of a rectangular prism. “The volume of a rectangular prism is 800 cubic inches. The length of the prism is 30 inches. The height is 13 inches. Which equation can be used to find the width of the prism? A. 800 = 30 x w x 13, B. 800 - w = 30 + 13, C. 800 = w(30 + 13), D. 800 - 30 - 13 = w.”

• In Unit 9, Add and Subtract Fractions, Performance Task, Valentina’s Celebration!, supports the full intent of 5.NF.2 (Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem), and MP4 (Model with mathematics) as students show two different ways to add mixed numbers. “Valentina’s Celebration! Valentina has several things to do before her party. Part A: Valentina drives $4\frac{2}{3}$ miles to the bakery to pick up muffins. Valentina then drives $5\frac{1}{2}$ miles to the grocery store to pick up some snacks for the party. How many miles did Valentine drive? Show two different ways to add.”

• Unit 13, Geometry, Unit Assessment, Form A, Item 5, supports the full intent of 5.G.2 (Represent real world and mathematical problems by graphing in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.), and MP6 (Attend to precision) as students interpret points on a coordinate plane to calculate accurately. “How many more printers were sold on Day 4 than on Day 3? A. 15 printers, B. 25 printers, C. 30 printers, D. 45 printers.”

• Summative Assessment, Item Analysis, Item 5 supports the full intent of 5.NBT.7 (Add, subtract, multiply and divide decimals to hundredths, using concrete models or drawings...), and and MP2 (Reason abstractly and quantitatively) as students consider the units involved in the problem and attend to the meaning of the quantities. “What is the best estimate of the product of 0.62 x 0.38. A. 0.18, B. 0.24, C. 1.8, D. 2.4.”

The materials reviewed for Reveal Math Grade 5 meet expectations for providing strategies and supports for students in special populations to support their regular and active participation in learning grade-level mathematics.

There are multiple locations of supports for students in special populations at the unit and lesson level. These supports are specifically aligned to lessons and standards, making them engaging in a variety of ways. They also scaffold up to the learning instead of simplifying or lowering expectations.

The Implementation Guide, Support for English Learners, identifies three features at the Unit level:

• “The Math Language Development feature offers insights into one of the four areas of language competence - reading, writing, listening, and speaking - strategies to build students’ proficiency with language.”

• The English Language Learner feature provides an overview of the lesson-level support.”  

• The Math Language Routines feature consists of a listing of the Math Language Routines found in each lesson of the unit.” 

The Implementation Guide, Support for English Learners, also identifies three features at the Lesson level:

• Language Objectives: “In addition to a content objective, each lesson has a language objective that identifies a linguistic focus for the lesson for English Learners. The language objective also identifies the Math Language Routines for the Lesson.”

• English Learner Scaffolds: “English Learner Scaffolds provide teachers with scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, Developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language Development Standards. With these three levels, teachers can scaffold instruction to the appropriate level of language proficiency for their students.”  

• Math Language Routines: “Each lesson has at least one Math Language Routine specifically designed to engage English Learners in math and language.”  

The Implementation Guide, Differentiation Resources, provides a variety of small group activities and resources to support differentiation to sufficiently engage students in grade level mathematics. Examples include: 

• Reinforce Understanding: “These teacher-facilitated small group activities are designed to revisit lesson concepts for students who may need additional instruction.”  

• Build Proficiency: “Students can work in pairs or small groups on the print-based Game Station activities, written by Dr. Nicki Newton, or they can opt to play a game in the Digital Station that helps build fluency.”  

• Extend Thinking: “The Application Station tasks offer non-routine problems for students to work on in pairs or small groups.”   

The Implementation Guide, Differentiation Resources, provides a variety of independent activities and resources to support differentiation to sufficiently engage students in grade level mathematics. Examples include:

• Reinforce Understanding: “Students in need of additional instruction on the lesson concepts can complete either the Take Another Look mini-lessons, which are digital activities, or the print-based Reinforce Understanding activity master.”

• Build Proficiency: “Additional Practice and Spiral Review assignments can be completed in either print or digital environment. The digital assignments include learning aids that students can access as they work through the assignment. The digital assignments are also auto-scored to give students immediate feedback on their work.”  

• Extend Thinking: “The STEM Adventures and Websketch activities powered by Geometer’s Sketchpad offer students opportunities to solve non-routine problems in a digital environment. The print-based Extend Thinking activity master offers an enrichment or extension activity.”  

The Teacher Edition and Implementation Guide provide overarching guidance for teachers on how to use the supports provided within the program. Examples include:

• Teacher Edition, Volume 1, Lesson Model: Differentiate, for every lesson, there are multiple options for teachers to choose to support student learning. Based on data from Exit Tickets, students can reinforce lesson skills with “Reinforce Understanding” opportunities, practice their learning with “Build Proficiency” opportunities, or extend and apply their learning with “Extend Thinking” opportunities. Within each of these opportunities, there are options of workstations, online activities and independent practice for teachers to elect to use. 

• Implementation Guide, Targeted Intervention, “Targeted intervention resources are available to assign students based on their performance on all Unit Readiness Diagnostics and Unit Assessments. The Item Analysis table lists the appropriate resources for the identified concept or skill gaps. Intervention resources can be found in the Teacher Center in both the Unit Overview and Unit Review and Assess sections.”  The Item Analysis can be found in the Teacher Edition. Intervention resources include Guided Support, “Guided Support provides a teacher-facilitated small group mini-lesson that uses concrete modeling and discussion to build conceptual understanding” and Skills Support, “Skills Support are skills-based practice sheets that offer targeted practice of previously taught items.”  Both of these can be located in the Digital Teacher Center.

The materials reviewed for Reveal Math Grade 5 meet expectations for providing extensions and/or opportunities for students to engage with grade-level mathematics at higher levels of complexity.

Each unit opens with an Ignite! activity that poses an interesting problem or puzzle to activate prior knowledge and spark students’ curiosity around the mathematics for the unit. In the Digital Teacher Center, “What are Ignite! Activities?” video, contributing author Raj Shah, Ph.D., explains, “An Ignite! Activity is an opportunity to build the culture of your classroom around problem-solving, exploration, discovery and curiosity.” The activity gives teachers, “the opportunity to see what the students can do on their own, without having to pre-teach them anything.” This provides an opportunity for advanced students to bring prior knowledge and their own abilities to make insightful observations. 

The Teacher Edition, Unit Resources At-A-Glance page includes a Workstations table which, “offers rich and varied resources that teachers can use to differentiate and enrich students’ instructional experiences with the unit content. The table presents an overview of the resources available for the unit with recommendations for when to use.” This table includes Games Station, Digital Station, and Application Station. 

Within each lesson, there are opportunities for students to engage in extension activities and questions of a higher level of complexity. The Practice & Reflect, On My Own section of the lesson provides an Item Analysis table showing the aspect of rigor and DOK level of each item. The Exit Ticket at the end of each lesson provides differentiation that includes extension through a variety of activities.

Additionally, there are no instances of advanced students doing more assignments than their classmates.

The materials reviewed for Reveal Math Grade 5 meet expectations for providing strategies and supports for students who read, write, and/or speak in a language other than English to regularly participate in learning grade-level mathematics.

The materials provide strategies for all students to foster their regular and active participation in learning mathematics, as well as, specific supports for English Learners. 

In the Implementation Guide, Support for English Learners, Unit-level support, “At the unit level are three features that provide support for teachers as they prepare to teach English Learners. The Math Language Development feature offers insights into one of the four areas of language competence - reading, writing, listening, and speaking - and strategies to build students’ proficiency with language. The English Language Learner feature provides an overview of lesson-level support. The Math Language Routines feature consists of a listing of the Math Language Routines found in each lesson of the unit.” The Unit Overview also includes a Language of Math section highlighting key vocabulary from the unit. These sections provide an overview of the strategies present within the unit ,and give guidance as to possible misconceptions or challenges that EL students may face with language demands. Included within the Unit Review is a Vocabulary Review that includes an Item Analysis for each item as well as what lesson/s the term was found in.  

At the lesson level, there are supports to engage ELs in grade-level content and develop knowledge of the subject matter. These involve oral language development and reading and writing activities. The Teacher Edition and Implementation Guide outline these features. Examples include:

• Language Objective, “In addition to a content objective, each lesson has a language objective that identifies a linguistic focus of the lesson for English Learners. The language objective also identifies the Math Language Routine of the lesson.”

• Math Language Routine, “Each lesson has at least one Math Language Routine specifically designed to engage English Learners in math and language.” Math Language Routines (MLR), listed and described in the Implementation Guide include: Stronger and Clearer Each Time, Collect and Display, Critique, Correct, and Clarify, Information Gap, Co-Craft Questions and Problems, Three Reads, Compare and Connect, Discussion Supports.

• English Learner Scaffolds, “English Learner Scaffolds provide teachers with scaffolded instruction to help students make meaning of math vocabulary, ideas, and concepts in context. The three levels of scaffolding within each lesson - Entering/Emerging, developing/Expanding, and Bridging/Reaching are based on the 5 proficiency levels of the WIDA English Language development Standards. With these three levels, teachers can scaffold instruction to the appropriate level of language proficiency of their students.”

• Language of Math, ”The Language of Math feature promotes the development of key vocabulary terms that support how we talk about and think about math in the context of the lesson content.”

• Number Routines such as “Would You Rather?” or “Math Pictures” and Sense-Making Routines such as “Notice and Wonder” or “Which Doesn’t Belong?” provide opportunities to develop and strengthen number sense and problem solving through discussion or written responses.

Most materials are available in Spanish such as the Student Edition, Student Practice Book (print), Student eBook, Math Replay Videos, eGlossary, and Family Letter (digital).

The materials reviewed for Reveal Math Grade 5 meet expectations for providing manipulatives, both virtual and physical, that are accurate representations of the mathematical objects they represent, and when appropriate, are connected to written methods.

Physical manipulatives needed for each unit and lesson can be found in the Teacher Edition, Unit Planner, at the beginning of each unit under “Materials to Gather”. Each lesson also identifies needed materials in the “Materials” section on the first page of each lesson.

Virtual manipulatives can be found online under “e-Toolkit”. Manipulatives are used throughout the program to help students develop a concept or explain their thinking. They are used to develop conceptual understanding and connect concrete representations to a written method.